Jean-Francois Le Gall (University of Paris)

	The Continuous Limit of Large Random Planar Graphs, I & II

Abstract

Planar maps are graphs embedded in the plane, considered up to continuous deformation. 
They have been studied extensively in combinatorics, and they also have significant
geometrical applications. Random planar maps have been used in theoretical physics, where
they serve as models of random geometry. Our goal is to discuss the convergence in 
distribution of rescaled random planar maps viewed as random metric spaces. More precisely, 
we consider a random planar map M(n) which is uniformly distributed over the set of all 
planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with 
the graph distance rescaled by the factor n to the power -1/4. We then discuss the convergence 
in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the 
Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm 
in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, 
we first establish a compactness result showing that a limit exists along a suitable subsequence. 
We then prove that this limit can be written as a quotient space of the so-called Continuum Random 
Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels 
attached to the vertices of the CRT. This limiting random metric space, which is called the Brownian 
map, can be viewed as a "Brownian surface" in the same sense as Brownian motion is the limit of 
rescaled discrete paths. We show that the Brownian map is almost surely homeomorphic to the two-
dimensional sphere, although it has Hausdorff dimension 4. Furthermore, we are able to give a 
complete description of the geodesics from a distinguished point (the root) of the Brownian map, 
and in particular of those points which are connected by more than one geodesic to the root. As 
a key tool, we use bijections between planar maps and various classes of labeled trees. 


References (available on http://www.dma.ens.fr/users/legall/ ) 

J.F. Le Gall. The topological structure of scaling limits of large planar maps. 
Invent. Math. 169, 621-670 (2007). 

J.F. Le Gall, F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 
2-sphere. Geometr. Funct. Analysis, to appear. 

J.F. Le Gall. Geodesics in large random planar maps. Preprint (2008)

Alexei Borodin (Caltech) 
      Growth of Random Surfaces
          

Abstract:  

Savas Dayanik (Princeton)

        Sequential Change Detection and Identification Problems for Stochastic Processes

Abstract: 

The characteristics of the stochastic models used to describe financial asset prices, hospital 
emergency room visits, computer network traffic are often expected to change at an unknown and 
unobservable time in the future, due to a shift in the economic cycle, a disease outbreak, or a 
cyber-attack. To hedge against the emerging new investment risks, protect the public health, 
prevent unauthorized access to computer network resources on a timely basis, one faces the problem 
of detecting those disorder times as quickly as possible while keeping the number of false alarms 
at acceptable low levels. We will show how these important problems can be formulated and solved 
in a Bayesian setting when the underlying stochastic model is a Levy process.